Abstract

In this thesis, we study two types of reaction-diffusion systems which have direct applications in understanding wide range of phenomena in chemical reaction, biological pattern formation and theoretical ecology. The first part of this thesis is on propagating traveling waves in a class of reaction-diffusion systems which model isothermal autocatalytic chemical reactions as well as microbial growth and competition in a flow reactor. In the context of isothermal autocatalytic systems, two different cases will be studied. The first is autocatalytic chemical reaction of order $m$ without decay. The second is chemical reaction of order $m$ with a decay of order $l$, where $m$ and $l$ are positive integers and $m>l\ge1$. A typical system is $A + 2B \rightarrow3B$ and $B\rightarrow C$ involving three chemical species, a reactant A and an auto-catalyst B and C an inert chemical species. We use numerical computation to give more accurate estimates on minimum speed of traveling waves for autocatalytic reaction without decay, providing useful insight in the study of stability of traveling waves. For autocatalytic reaction of order $m = 2$ with linear decay $l = 1$, which has a particular important role in biological pattern formation, it is shown numerically that there exist multiple traveling waves with 1, 2 and 3 peaks with certain choices of parameters. The second part of this thesis is on the global stability of diffusive predator-prey system of Leslie Type and Holling-Tanner Type in a bounded domain $\Omega\subset R^N$ with no-flux boundary condition. By using a new approach, we establish much improved global asymptotic stability of a unique positive equilibrium solution. We also show the result can be extended to more general type of systems with heterogeneous environment and/or other kind of kinetic terms.

Graduation Date

2016

Semester

Fall

Advisor

Qi, Yuanwei

Degree

Doctor of Philosophy (Ph.D.)

College

College of Sciences

Department

Mathematics

Degree Program

Mathematics

Format

application/pdf

Identifier

CFE0006519

URL

http://purl.fcla.edu/fcla/etd/CFE0006519

Language

English

Release Date

December 2016

Length of Campus-only Access

None

Access Status

Doctoral Dissertation (Open Access)

Included in

Mathematics Commons

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